# Varieties of Cubical Sets

## TL;DR

This paper introduces various notions of cubical sets based on algebraic theories, proving their effectiveness in modeling classical homotopy theory and identifying strict test categories where products align with homotopy types.

## Contribution

It defines new classes of cubical sets using substructural algebraic theories and establishes their properties as test categories for homotopy modeling.

## Key findings

- All defined sites are test categories modeling homotopy theory
- Some sites are strict test categories with product-homotopy type correspondence
- Clarifies the algebraic structures underlying cubical sets in homotopy theory

## Abstract

We define a variety of notions of cubical sets, based on sites organized using substructural algebraic theories presenting PRO(P)s or Lawvere theories. We prove that all our sites are test categories in the sense of Grothendieck, meaning that the corresponding presheaf categories of cubical sets model classical homotopy theory. We delineate exactly which ones are even strict test categories, meaning that products of cubical sets correspond to products of homotopy types.